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Deutsch-Yozhi Algorithm

Lecture



The Deutsch – Jozy algorithm (also referred to as the Deutsch – Josy algorithm) is a quantum algorithm proposed by David Deutsch and Richard Jozhey [en] in 1992, and one of the first examples of algorithms designed to run on quantum computers. Thanks to the use of the quantum entanglement phenomenon and the principle of superposition, these algorithms have a significant increase in execution speed compared to the corresponding classical algorithms.

  Deutsch-Yozhi Algorithm

The task of Deutsch - Yozhi is to determine whether a function is a binary variable   Deutsch-Yozhi Algorithm constant (takes either the value 0 or 1 for any arguments) or balanced (for the half of the definition area takes the value 0, for the other half 1). It is considered a priori known that the function is either constant or balanced.

To solve this problem, the classical deterministic algorithm needs to be produced   Deutsch-Yozhi Algorithm computing the function f in the worst case. The classical probabilistic algorithm takes less time to give the correct answer with a high probability. But in any case, to get the right answer with a single probability will be required   Deutsch-Yozhi Algorithm calculations The algorithm gives the correct answer, once applying the phase request corresponding to the given function f .

If the function   Deutsch-Yozhi Algorithm unbalanced, the algorithm can give the answer "constant" with a certain probability, and the greater the difference between the number of "0" and "1", the greater will be this probability [1] .

The algorithm of Deutsche-Jozi is based on a similar algorithm developed by David Deutsch in 1985, which is a special case of the first one. In this algorithm, the function   Deutsch-Yozhi Algorithm was a function of one variable, in contrast to the function of many variables   Deutsch-Yozhi Algorithm used in a later algorithm.

Algorithm

At the input of the algorithm there is a boolean function   Deutsch-Yozhi Algorithm from   Deutsch-Yozhi Algorithm Boolean variables. The algorithm is an application to the zero vector.   Deutsch-Yozhi Algorithm operator   Deutsch-Yozhi Algorithm and register state measurement. If all bits of the register are 0, then the value of the function   Deutsch-Yozhi Algorithm does not depend on   Deutsch-Yozhi Algorithm otherwise the function is balanced.

Here   Deutsch-Yozhi Algorithm - Hadamard transform:   Deutsch-Yozhi Algorithm

  Deutsch-Yozhi Algorithm - phase request that inverts the phase for register states corresponding to the function units   Deutsch-Yozhi Algorithm and does not change the state corresponding to the zeros of the function:   Deutsch-Yozhi Algorithm [2]

The operation of the algorithm on the example of the task Deutsch

The input to the algorithm is a Boolean function of one variable.   Deutsch-Yozhi Algorithm . In total there are 4 such functions [2] :

No f (0) f (1)
one 0 0   Deutsch-Yozhi Algorithm
2 one one   Deutsch-Yozhi Algorithm
3 0 one   Deutsch-Yozhi Algorithm
four one 0   Deutsch-Yozhi Algorithm

Functions 1 and 2 are called constant, and functions 3 and 4 are balanced.

At the first step, we set the zero state to the qubit:   Deutsch-Yozhi Algorithm

Applying the Hadamard transform we get   Deutsch-Yozhi Algorithm . In principle, it would be possible to immediately assign a state to a qubit, but it is technically easier to first set the zero state and then convert it using unitary transformations to the desired form.

Applying phase request   Deutsch-Yozhi Algorithm to our function   Deutsch-Yozhi Algorithm , we get the following state:   Deutsch-Yozhi Algorithm

The second Hadamard transform leads to the following state:   Deutsch-Yozhi Algorithm

When measuring the state of a qubit, we obtain 0 for constant functions and 1 for balanced functions. This can be seen by substituting all sorts of functions f (x) into the expression for the qubit state:

No f (0) f (1) Qubit state   Deutsch-Yozhi Algorithm Probability of receiving 0 Probability of receiving 1
one   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm
2   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm
3   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm
four   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm   Deutsch-Yozhi Algorithm

Notes

  1. M. Sluggish, Quantum Algorithms, Lecture 2.
  2. ↑ Go to: 1 2 Quantum algorithms: possibilities and limitations. M. Vyaly, Lecture 2, pp. 12-13 (St. Petersburg, spring 2011)
created: 2016-04-02
updated: 2021-03-13
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Quantum informatics

Terms: Quantum informatics